Projection and Clipping網媒所林宏祥 2012.10.02

Overview

Orthogonal Projection

-Z

X

xcameraxviewport

cameraviewport xx

Perspective Projection

-Z

X

xcamera

xviewport

d

z

dz

xx cameraviewport

View Volume—orthogonal projection

Near Clipping Plane

Far Clipping PlaneLeft Clipping Plane

Right Clipping Plane

Bottom Clipping Plane

Top Clipping Plane

x

y

-z

View Volume—Perspective projection model

Far Clipping Plane

Near Clipping Plane

Left Clipping Plane

Right Clipping Plane

Bottom Clipping Plane

Top Clipping Plane

x

y

-z

z

x

Clipping Cuboid: transformed volume after projection

Yviewport coordinates

Implementation behind OpenGL Functions….

In OpenGL, the clipping cuboid is fixed.(Since the screen size is fixed)

z

x

Clipping Cuboid: enclosed by: xviewport = 1,xviewport=-1,yviewport=1,yviewport=-1,zviewport =1,zviewport=-1

Y

The openGL functions ensure the view volume fits clipping cuboid.

glOrtho(left, right, bottom, top, nearVal, farVal)

farValleft

right

bottom

top

nearVal

glOrtho(left, right, bottom, top, nearVal, farVal)

1

1000

200

020

002

camera

camera

camera

zyx

nearValfarValnearValfarVal

nearValfarVal

bottomtopbottomtop

bottomtop

leftrightleftright

leftright

Orthogonal Projection

-Z

X

xcameraxviewport

cameraviewport xx

some scaling

A linear mapping for orthogonal projection

A canonical view..

xviewport0

1-1

xcamera

rightleft

Clipping cuboid Viewing Volume

2rightleft

1)(10x

leftright2

leftrightxviewport

camera

Same mapping ways in y direction and z direction..

gluPerspective(fovy, aspect, zNear, zFar)

Far Clipping Plane

Near Clipping Plane

Left Clipping Plane

Right Clipping Plane

Bottom Clipping Plane

Top Clipping Plane

1

0100

200

100

100

camera

camera

camera

zyx

zFarzNearzNearzFar

zFarzNearzNearzFar

faspectf

gluPerspective(fovy, aspect, zNear, zFar)

)2/tan(1fovy

f

Perspective Projection

-Z

X

xcamera

xviewport

d

z

= 1

some scaling

-Z

X

xcamera

xviewport

d

z

dz

xx cameraviewport

A canonical view..

-Z

-zNear

-zFar

ycamera

zcamerafovy

Y

f1

zy)

2fovytan(

camera

camera

-Z

-zNear

-zFar

ycamera

zcamerafovy

Y

After projection, the boundaries should map to 1, -1)

fz-(

yycamera

cameraviewport

-Z

-zNear

-zFar

ycamera

zcamerafovy

Y

Similarly in x, but consider the aspect.

)f

z-(aspect

xxcamera

cameraviewport

)f

z-(

yycamera

cameraviewport

Mapping on z: To present more details in closer objects, the viewport coordinate is inverse proportional to camera coordinates.

cz

m

camera

viewportz

cameraz1

viewportz

cz

m

camera

viewportz

-zFar is mapping to 1, -zNear is mapping to -1, then

czFar-m1

czNear-

m1 zFar-zNear

zNearzFar c ,zFar)-(zNear

zNearzFar2m

zFar-zNearzNearzFar

)zFar)(-z-(zNearzNearzFar2z

cameraviewport

)f

z-(

yycamera

cameraviewport

)f

z-(aspect

xxcamera

cameraviewport

1

0100

200

100

100

camera

camera

camera

zyx

zFarzNearzNearzFar

zFarzNearzNearzFar

faspectf

Remember perspective projection?(xcamera, ycamera coordinates will be divided by zcamera)

Perspective projection <--> maintaining homogeneous coordinates

wzyx

1wzwywx

In the clipping part of your homework..

You need to create your own clipping cuboid which is different from openGL clipping cuboid. NOT modify the viewing volume.

(eg: xviewport: [-0.5,0.5], yviewport:[-0.5,0.5], zviewport:[-0.5, 0.5])

You need to adjust the coordinates in glVertex3f()

You need to calculate intersectionIn 2D example,

screen

One capable way to handle it: Cohen-Sutherland Algorithm

Any Question?